Brief History :

Brief History Classical logic of Aristotle: Law of Bivalence “Every proposition is either True or False(no middle)”
Jan Lukasiewicz proposed three-valued logic : True, False and Possible
Finally Lofti Zadeh published his paper on fuzzy logic-a part of set theory that operated over the range [0.0-1.0] 3

What is Fuzzy Logic? :

What is Fuzzy Logic? Fuzzy logic is a superset of Boolean (conventional) logic that handles the concept of partial truth, which is truth values between "completely true" and "completely false”.
Fuzzy logic is multivalued. It deals with degrees of membership and degrees of truth.
Fuzzy logic uses the continuum of logical values between 0 (completely false) and 1 (completely true). Boolean
(crisp) Fuzzy 4

:

For example, let a 100 ml glass contain 30 ml of water. Then we may consider two concepts: Empty and Full.
In boolean logic there are two options for answer i.e. either the glass is half full or glass is half empty. 100 ml 30 ml In fuzzy concept one might define the glass as being 0.7 empty and 0.3 full. 5

Crisp Set and Fuzzy Set :

Crisp Set and Fuzzy Set 6 μ a(x)={ 1 if element x belongs to the set A
0 otherwise
} Classical set theory enumerates all element using A={a1,a2,a3,a4…,an} Set A can be represented by Characteristic function A fuzzy set can be represented by:
A={{ x, A(x) }}
where, A(x) is the membership grade of a element x in fuzzy set
SMALL={{1,1},{2,1},{3,0.9},{4,0.6},{5,0.4},{6,0.3},{7,0.2},{8,0.1},{9,0},{10,0},{11,0},{12,0}} In fuzzy set theory elements have varying degrees of membership Example: Consider space X consisting of natural number<=12
Prime={x contained in X | x is prime number={2,3,5,7,11}

Fuzzy Vs. Crisp Set :

Fuzzy Vs. Crisp Set A A’ a a b b c Fuzzy set Crisp set a: member of crisp set A
b: not a member of set A a: full member of fuzzy set A’
b: not a member of set A’
c:partial member of set A’ 7

Fuzzy Vs. Crisp Set :

Crisp set Fuzzy Vs. Crisp Set Fuzzy set 8

:

Features of a membership function core support boundary 1 0 μ (x) x Core: region characterized by full membership in set A’ i.e. μ (x)=1.
Support: region characterized by nonzero membership in set A’ i.e. μ(x) >0.
Boundary: region characterized by partial membership in set A’ i.e. 0< μ (x) <1 9 A membership function is a mathematical function which defines the degree of an element's membership in a fuzzy set.

Membership Functions :

Fuzzy Logic Vs Probability :

Fuzzy Logic Vs Probability Both operate over the same numeric range and at first glance both have similar values:0.0 representing false(or non-membership) and 1.0 representing true.
In terms of probability, the natural language statement would be ”there is an 80% chance that Jane is old.”
While the fuzzy terminology corresponds to “Jane’s degree of membership within the set of old people is 0.80.’
Fuzzy logic uses truth degrees as a mathematical model of the vagueness phenomenon while probability is a mathematical model of ignorance. 11

Why use Fuzzy Logic? :

Why use Fuzzy Logic? Fuzzy logic is flexible.
Fuzzy logic is conceptually easy to understand.
Fuzzy logic is tolerant of imprecise data.
Fuzzy logic is based on natural language. 12

Drawbacks :

Fuzzy logic is not always accurate. The results are perceived as
a guess, so it may not be as widely trusted .
Requires tuning of membership functions which is difficult to
estimate.
Fuzzy Logic control may not scale well to large or complex
problems
Fuzzy logic can be easily confused with probability theory, and
the terms used interchangeably. While they are similar concepts,
they do not say the same things. Drawbacks 23

Conclusion :

Fuzzy Logic provides way to calculate with imprecision and
vagueness.
Fuzzy Logic can be used to represent some kinds of human
expertise .
The control stability, reliability, efficiency, and durability of fuzzy
logic makes it popular.
The speed and complexity of application production would not be
possible without systems like fuzzy logic. Conclusion 24